In Exercises 1 through 6, determine whether the given map o is a homomorphism. [Hint: The straightforward way to proceed

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In Exercises 1 Through 6 Determine Whether The Given Map O Is A Homomorphism Hint The Straightforward Way To Proceed 1 (36.33 KiB) Viewed 45 times

In Exercises 1 through 6, determine whether the given map o is a homomorphism. [Hint: The straightforward way to proceed is to check whether o(ab) = o(a)o(b) for all a b in the domain of o. However if we should happen to notice that [{e}] is not a subgroup whose left and right cosets coincide, or that does not satisfy the properties of Exercise 44 or 45 for finite groups, then we can say at once that is not a homomorphism.] 1 Let ø: Z → R under addition be given by o(n) = 2n. 2 Leto: R* → R* under multiplication be given by p(x) = |x³|. 3 Let : Z10 → Z₂ be given by p(x) = the remainder of z when divided by 2, as in the division algorithm. 4 Let : Z15 → Z₂ be given by o(z) = the remainder of z when divided by 2, as in the division algorithm. 5 Let G be any group and let : G→G be given by o(g) = g¹ for g € G. 6 Let M₁, be the additive group of all n x n matrices with real entries, and let R be the additive group of real numbers. Let o(A) = det(A), the determinant of A, for A € M₁-